Abstract:
An asymptotic expansion for inverse moments of positive binomial and Poisson distributions is derived. The expansion coefficients of the asymptotic series are given by the positive central moments of the distribution. Compared to previous results, a single expansion formula covers all (also non-integer) inverse moments. In addition, the approach can be generalized to other positive distributions.

Abstract:
An asymptotic expansion for inverse moments of positive binomial and Poisson distributions is derived. The expansion coefficients of the asymptotic series are given by the positive central moments of the distribution. Compared to previous results, a single expansion formula covers all (also non-integer) inverse moments. In addition, the approach can be generalized to other positive distributions.

Abstract:
A study of the first four moments (mean, variance, skewness, and kurtosis) and their products ($\kappa\sigma^{2}$ and $S\sigma$) of the net-charge and net-proton distributions in Au+Au collisions at $\sqrt{\rm s_{NN}}$ = 7.7-200 GeV from HIJING simulations has been carried out. The skewness and kurtosis and the collision volume independent products $\kappa\sigma^{2}$ and $S\sigma$ have been proposed as sensitive probes for identifying the presence of a QCD critical point. A discrete probability distribution that effectively describes the separate positively and negatively charged particle (or proton and anti-proton) multiplicity distributions is the negative binomial (or binomial) distribution (NBD/BD). The NBD/BD has been used to characterize particle production in high-energy particle and nuclear physics. Their application to the higher moments of the net-charge and net-proton distributions is examined. Differences between $\kappa\sigma^{2}$ and a statistical Poisson assumption of a factor of four (for net-charge) and 40% (for net-protons) can be accounted for by the NBD/BD. This is the first application of the properties of the NBD/BD to describe the behavior of the higher moments of net-charge and net-proton distributions in nucleus-nucleus collisions.

Abstract:
A family of generalized binomial probability distributions attached to Landau levels on the Riemann sphere is introduced by constructing a kind of generalized coherent states. Their main statistical parameters are obtained explicitly. As an application, photon number statistics related to coherent states under consideration are discussed.

Abstract:
Following the relationship between probability distribution and coherent states, for example the well known Poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the $su(2)$ coherent states, we propose ``interpretation'' of $su(1,1)$ and $su(r,1)$ coherent states ``in terms of probability theory''. They will be called the ``negative binomial'' (``multinomial'') ``states'' which correspond to the ``negative'' binomial (multinomial) distribution, the non-compact counterpart of the well known binomial (multinomial) distribution. Explicit forms of the negative binomial (multinomial) states are given in terms of various boson representations which are naturally related to the probability theory interpretation. Here we show fruitful interplay of probability theory, group theory and quantum theory.

Abstract:
We approximate the distribution of the sum of independent but not necessarily identically distributed Bernoulli random variables using a shifted binomial distribution where the three parameters (the number of trials, the probability of success, and the shift amount) are chosen to match up the first three moments of the two distributions. We give a bound on the approximation error in terms of the total variation metric using Stein's method. A numerical study is discussed that shows shifted binomial approximations typically are more accurate than Poisson or standard binomial approximations. The application of the approximation to solving a problem arising in Bayesian hierarchical modeling is also discussed.

Abstract:
For delta operator $aD-bD^{p+1}$ we find the corresponding polynomial sequence of binomial type and relations with Fuss numbers. In the case $D-\frac{1}{2}D^2$ we show that the corresponding Bessel-Carlitz polynomials are moments of the convolution semigroup of inverse Gaussian distributions. We also find probability distributions $\nu_{t}$, $t>0$, for which $\left\{y_{n}(t)\right\}$, the Bessel polynomials at $t$, is the moment sequence.

Abstract:
The nonnegativity of the determinant of the partial transpose of a two-qubit (4 x 4) density matrix is both a necessary and sufficient condition for its separability. While the determinant is restricted to the interval [0,1/256], the determinant of the partial transpose can range over [-1/16,1/256], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of the partial transpose over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of real two-qubit density matrices. Rational functions C_{2 j}(m), yielding the coefficients of the 2j-th power of even polynomials occurring at intermediate steps in our derivation of the m-th moment, emerge. These functions possess poles at finite series of consecutive half-integers (m=-3/2,-1/2,...,(2j-1)/2), and certain (trivial) roots at finite series of consecutive natural numbers (m=0, 1,...). Additionally, the (nontrivial) dominant roots of C_{2 j}(m) approach the same half-integer values (m = (2 j-1)/2, (2 j-3)/2,...), as j increases. The first two moments (mean and variance) found--when employed in the one-sided Chebyshev inequality--give an upper bound of 30397/34749 = 0.874759 on the separability probability of real two-qubit density matrices. We are able to report general formulas for the m-th moment of the Hilbert-Schmidt probability distribution of the density matrix determinant over [0,1/256], in the real, complex and quaternionic two-qubit cases.

Abstract:
The binomial outcome data are widely encountered in many real world applications. The Binomial distribution often fails to model the binomial outcomes since the variance of the observed binomial outcome data exceeds the nominal Binomial distribution variance, a phenomenon known as overdispersion. One way of handling overdispersion is modeling the success probability of the Binomial distribution using a continuous distribution defined on the standard unit interval. The resultant general class of univariate discrete distributions is known as the class of Binomial mixture distributions. The Beta-Binomial (BB) distribution is a prominent member of this class of distributions. The Kumaraswamy-Binomial (KB) distribution is another recent member of this class. In this paper we focus the emphasis on the McDonald's Generalized Beta distribution of the first kind as the mixing distribution and introduce a new Binomial mixture distribution called the McDonald Generalized Beta-Binomial distribution(McGBB). Some theoretical properties of McGBB are discussed. The parameters of the McGBB distribution are estimated via maximum likelihood estimation technique. A real world dataset is modeled by using the new McGBB mixture distribution, and it is shown that this model gives better fit than its nested models. Finally, an extended simulation study is presented to compare the McGBB distribution with its nested distributions in handling overdispersed binomial outcome data.

Abstract:
A Poisson Binomial distribution over $n$ variables is the distribution of the sum of $n$ independent Bernoullis. We provide a sample near-optimal algorithm for testing whether a distribution $P$ supported on $\{0,...,n\}$ to which we have sample access is a Poisson Binomial distribution, or far from all Poisson Binomial distributions. The sample complexity of our algorithm is $O(n^{1/4})$ to which we provide a matching lower bound. We note that our sample complexity improves quadratically upon that of the naive "learn followed by tolerant-test" approach, while instance optimal identity testing [VV14] is not applicable since we are looking to simultaneously test against a whole family of distributions.